IP Maths Notes (Lower Sec, Year 1-2): 08) Trigonometry in Right Triangles

Study guide23 Nov 2025, 00:00 Z

Apply sine, cosine, and tangent to height-distance problems, bearings, and angles of elevation with calculator and non-calculator techniques.

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Q: What does IP Maths Notes (Lower Sec, Year 1-2): 08) Trigonometry in Right Triangles cover?
A: Apply sine, cosine, and tangent to height-distance problems, bearings, and angles of elevation with calculator and non-calculator techniques.

The core idea is simple: Label opposite, adjacent, and hypotenuse before choosing a trig ratio.

Use it as a working check: Use sine, cosine, or tangent when the triangle is right-angled. Use inverse trig when the angle is unknown, and always draw elevation, depression, or bearing diagrams first.

Then go one layer deeper: Work through the side, angle, drone, and bearing examples to practise translating a word problem into a right triangle before calculating.

Trigonometry unlocks height and distance tasks across science and geography contexts. Become fluent with ratios, inverse functions, and diagram interpretation.

These notes align with MOE Lower Secondary Mathematics syllabus used in IP pathways (aligned to O-Level Mathematics 4052 foundations).

Status: MOE Lower Secondary Mathematics syllabus (latest release) checked 2025-11-30 - scope unchanged; remains the reference for these lower-sec notes.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

Learning targets

Recall primary trigonometric ratios: $\sin \theta = \frac{\text{opp}}{\text{hyp}}$

Reviewed by

Marcus Pang·Managing Director (Maths)

Sources

SEAB - Mathematics (4052) GCE O-Level 2026 syllabus (PDF)

Step	What to mark	Check
1	The right angle.	The side opposite it is always the hypotenuse.
2	The angle you are using, $\theta$ .	Opposite and adjacent depend on this angle, not on the diagram's orientation.
3	The side directly across from $\theta$ .	This is the opposite side.
4	The side touching $\theta$ that is not the hypotenuse.	This is the adjacent side.

Known and unknown sides	Ratio to start with
Opposite and hypotenuse	$\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$
Adjacent and hypotenuse	$\cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$
Opposite and adjacent	$\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}$

Triangle	Side ratio to remember	If the focus angle is...	Example ratio	Common trap
30-60-90	short : long : hypotenuse = $1 : \sqrt{3} : 2$	$30^\circ$	$\sin 30^\circ = \frac{1}{2}$	Using the side opposite $60^\circ$ by mistake.
30-60-90	short : long : hypotenuse = $1 : \sqrt{3} : 2$	$60^\circ$
45-45-90	equal : equal : hypotenuse = $1 : 1 : \sqrt{2}$	$45^\circ$

Phrase in the question	Draw this first	Common trap
Angle of elevation	Horizontal eye-level line, then line of sight upwards.	Measuring the angle from the vertical.
Angle of depression	Horizontal eye-level line, then line of sight downwards.	Forgetting it equals the alternate interior angle at ground level when horizontals are parallel.
Bearing	North line at the starting point, then clockwise angle to the path.	Treating bearing as an ordinary angle inside the triangle before drawing north.
Height or depth	Vertical side of the right triangle.	Using sine automatically instead of checking which length is given.
Horizontal distance	Ground line or parallel horizontal line.	Confusing it with the slanted line of sight.

Step	What to define	Why it helps
1	Let the unknown distance from the nearer observer to the base be $x$ .	The farther observer's distance can be written using the given separation.
2	Write the other horizontal distance as $x + d$ or $d - x$ , depending on the diagram.	This keeps both right triangles tied to the same baseline.
3	Use $\tan(\text{angle}) = \dfrac{\text{height}}{\text{horizontal distance}}$ for each observer.	Both equations share the same vertical height.
4	Equate the two height expressions, then solve for $x$ before finding the height.	Solving the distance first avoids guessing which triangle gives the answer.

IP Maths Notes (Lower Sec, Year 1-2): 08) Trigonometry in Right Triangles

Learning targets

Sources

1 Ratio recap

Side-labelling map

Worked example - Finding a side

Worked example - Finding an angle

Worked example - Choosing the ratio from labels

2 Exact values and surds

Exact-value triangle checkpoint

3 Angle of elevation/depression

Worked example - Two-point observation

Two-observer height checkpoint

4 Bearings and composite paths

Bearing component checkpoint

Preview: sine and cosine rules

Practice Quiz

Try it yourself

Related Posts

Bearing type	East-west component	North-south component	Common trap
$\theta$ east of north, such as $\pu{040^\circ}$	$+d\sin\theta$	$+d\cos\theta$	Swapping sine and cosine because the angle is drawn near the horizontal.
$\theta$ west of north, such as $\pu{320^\circ}$	$-d\sin 40^\circ$	$+d\cos 40^\circ$
$\theta$ east of south, such as $\pu{140^\circ}$	$+d\sin 40^\circ$	$-d\cos 40^\circ$
Two-leg journey	Add all east-west components, then add all north-south components.	Use Pythagoras only after components are combined.	Finding each leg's displacement and adding lengths directly.